In the beginning,
y0, the husband invests D dollars in Stock No. 1.
At the end of the first year, y1, he has a profit of p+q
dollars.

He sells
Stock No. 1 and pays a tax of q dollars on the year's profit. This leaves
him D+p dollars—his original investment plus his after-tax profit.

He immediately
reinvests D+p dollars in Stock No. 2, which appreciates at exactly the
same rate as Stock No. 1. At the end of the second year, y2,
Stock No. 2 shows a profit of r+s+t+u dollars. The profit r+s
was earned on D, his original investment; the profit t+u was earned
on p, his after-tax profit on Stock No. 1.

He sells
Stock No. 2 at the end of the second year and pays his tax, which amounts to
s dollars on his profit r+s, and u dollars on his profit
t+u. After paying the tax, the husband has D+p+r+t dollars.

In the beginning,
y0, the wife—like her husband—invests D dollars in
Stock No. 1. And at the end of the first year, y1, she also
has a profit of p+q dollars. But unlike her husband, she does not sell;
she leaves her money invested in Stock No. 1. As a result she pays no taxes
at the end of year one.

By the end
of the second year, y2, she has an additional profit of r+s+t+u+v.
The profit r+s was earned on D, her original investment; the profit
t+u+v was earned on p+q, her first-year pre-tax profit.

At this
point, she does sell. She pays a tax of q dollars on p+q, her
first year profit, and she pays a tax of s+v on r+s+t+u+v, her
second year profit. After paying the tax q+s+v, she has D+p+r+t+u
dollars.

Act 2, Scene 1The Reckoning

The husband stacks his D+p+r+t dollars next to
his wife's D+p+r+t+u dollars and is dismayed to see that she has u
more dollars than he has.

"How
is this possible?" he exclaims. "Our investments grew at the
same rate, and our profits were taxed at the same rate."

"Simple,"
explains the wife. "By not paying the government q dollars in taxes
at the end of year one, I was able to earn an after-tax profit in year two on
the growth of q. My after-tax profit on that growth was u."

"I don't get it," says the husband.

"Okay,
try it this way." The wife starts over.

"In
year one you had a profit of p+q on Stock No. 1, but you sold it; so
you had to pay q in taxes. I, too, had a profit of p+q, but I
didn't sell; so I still had the government's as-yet-uncollected tax money, q,
working for me.

"In
year two you earned a profit on p, but I earned a profit on p+q.
Your profit on p was t+u, whereas my profit on p+q was
t+u+v. Your tax on t+u was u; that left you with t.

"My
tax on t+u+v was v, and that left me with t+u. My t+u
exceeds your t by u."

"Bummer,"
laments the husband.

Act 2, Scene 2
The Mysterious Stranger

Inside the
husband's head, wheels begin to turn. "Even though my wife and I invested
the same amount of money," he thinks, "and even though our investments
grew at the same rate, I started the second year with less money than
my wife—all because of the taxes on my first sale."

The wheels
pick up speed. "To stay even with my wife," he reasons, "I'll
need a higher growth rate on my investment. But how much higher?"

At that
very moment, the author of this essay, who has a direct pipeline to the husband's
thoughts, takes aim at the couple's doorbell and delivers a long volley of rapid
rings. With no little annoyance, the husband goes to the door, opens it slightly,
and peers through the crack. The author forces his right Rockport into the narrow
slit, gives a hardy shove, and barges in—uninvited.

The
husband and wife are aghast. But before they can so much as exhale, the intruder
launches a salvo of financial verbiage that reverberates through the house.

"How
much higher is a relative matter," he shouts, pulling a spreadsheet
out of his pocket. "It depends on the tax rate, the trading frequency,
the annualized yield on your wife's long-term investment, and the elapsed time
when she closes it out. Consider Table 1!" He waves Table 1 in the air,
but doesn't let them see it.

The wife, always a clever one, feigns interest and moves
toward him, grabbing a poker as she passes the fireplace.

"Don't club him till I hear what he has to say,"
mutters the husband, annoyance suddenly taking a backseat to greed.

"There are a few assumptions behind Table 1,"
confesses the author, who glimpses the approaching poker and quickly adopts
a deferential manner, "but they're reasonable."

"Like what?" asks the husband pointedly.

"Well,
the tax rate for one thing. The table assumes a combined federal and state
income tax rate of 28%—a 20% federal capital gains rate plus an 8% state income
tax rate. You can't argue with that assumption."

"No, I can't," admits the husband, calming
down a bit.

"That's
exactly what our tax is," the wife chimes in as she adjusts her grip on
the poker. "But I try to avoid tax."

"What else?" asks the husband in a nearly
normal tone.

"Your
trading frequency," replies the author. "The table assumes that you
hold each investment for exactly one year (just long enough to qualify for the
lower, long term capital gains rate) and that you immediately roll your after-tax
proceeds over into another stock—which is what you've been doing."

The wife nods in agreement and relaxes her grip.

"Using those assumptions, the table shows how much higher
your annualized rate of return must be for you to earn the same after-tax profit
as your 'buy-and-hold' spouse."

"But
we don't really know the annualized rate at which her investment will grow,"
complains the husband, his agitation on the rise again. "And we don't
know how long she will hold it."

"True
enough," agrees the author. "But this is a table. It shows a range
of growth rates and holding periods. You wife's situation is likely to fall
within this range, and . . ."

"Do you mean to tell me," the husband asks after
looking at the table, "that an 11% annualized return on a stock held for
a decade produces more after-tax profit than a 12% annualized return on a succession
of stocks traded yearly over the same period?"

"That's
right," answers the author. "And if a person holds that 11% stock
for two decades, it will be even harder for a yearly trader to come out
ahead. He'll need a return in excess of 13%."

"And
look at this!" The wife points to the last column and addresses her husband.
"If I luck out and snare a stock that appreciates 15% a year, you'll never
catch me."

"Well,
it would be very difficult," comments the author tactfully to the husband.
"If she can achieve 15%, an annual trader will have to do 2.1 points better
per year over ten years—and 3.4 points better per year over twenty years—just
to tie her."

"Do you think you can do that consistently?"
the wife asks her mate, giving him a playful nudge with her elbow.

"Probably
not," her husband acknowledges. "Not if these figures are correct.
But how do I know they are?"

"If
your wife will put down her poker, I'll prove it. But for your own safety,
you'd better take a seat. When I gave this explanation to an investment club,
people were dropping like hailstones."

Act 3, Scene 2
The Hailstorm

"Say,
you buy a stock," the author says to the wife as she and her husband settle
in. "And, say, its price appreciates a% annually."

"In
one year, your investment will be worth 100% of its original value plus
another a%. Since 100% equals 1, your investment's future value factor
for one year is 1+a%. In other words, if you multiply the number of
dollars you originally invested by the factor 1+a%, you'll get the value
of your investment one year into the future."

"Basic finance," says the
wife.

"Suppose
you hold your investment for y years. The future value factor for y
years is (1+a%)y. At the end of y years, your investment
will be worth (1+a%)y times its original value."

"That's y different (1+a%), all multiplied
together," mimes the wife, who has a knack for winning at charades: "If
y happens to be 3 years, it's (1+a%) * (1+a%) * (1+a%)."

The husband
shifts in his LA-Z-Boy. "Let's hear about profit."

"At
the end of y years, your profit will equal your original investment times
the factor [(1+a%)y -1]. That - 1 at the end is there
to subtract out the money you originally invested.

"And when you sell your stock, the profit will
be taxed at a rate of t%."

"I try to avoid tax," reaffirms the wife.

"The
percent of the profit remaining after taxes will be 1-t%. In
dollars, the after-tax profit will be your original investment times
the factor [(1+a%)y -1](1-t%)."

The author
presses on. "Let's assume that there is a second investor, an active trader.
This guy buys a stock, and its price appreciates b% over a year. After
one year, he sells it, pays his taxes, and reinvests his residual funds in another
stock. Say, he does this year after year."

"Like I do," says the husband.

"And,
say, each investment earns him the same annual return, b%. His after-tax
profit at the end of the first year is his initial investment times the factor
b%(1-t%). The after-tax money available for reinvestment at the start
of the second year is his initial investment times the factor [1+ b%(1-t%)].

"Now let's jump into the future."

"How far?" asks the husband, suppressing a yawn.

"How about y years?" suggests the wife.

"Fine,"
says the author. "At the end of y years, the total after-tax profit
on his original investment is the investment times the factor { [1+ b%(1-t%)]
y –1}.

The husband barely catches himself as his eyelids drop
and his chin plummets to his chest.

"We
almost there," responds the author. "And I think you'll like the
conclusion. It's the answer you were after when I rang the doorbell."

The author,
on his last lap to a spectacular finish, ignores her. "Let's recap. Two
investors start with the same amount of money. Both stay invested for y
years. What annualized rate of return, b%, does the active trader need
in order to earn the same after-tax profit as the long-term investor?